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Unformatted text preview: z 7 (1 + i ) z + 1. (b). Describe the image of S under the map z 7 e z . (c). Describe the image of S under the map z 7 Arg( z + i ). (Arg denotes the principal branch of the argument.) (d). Describe the images of S under the maps z 7 Arg ( z + i ), where Arg is any branch of the argument dened on S . 5. (15 points) Let f ( z ) = x 3 + i (1-y ) 3 (where z = x + iy as usual). Show that the only point where f is dierentiable is z = i . 6. (15 points) Let f ( z ) = f ( x + iy ) = u ( x,y )+ iv ( x,y ) is entire and that the rst-order partial derivatives of u and v are continuous. Let h ( z ) = f ( z ). Show that h ( z ) is entire....
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- Fall '07